how do i solve:
log4x^3 + log2x^1/2 = 8
please help me
You should use parentheses to indicate just what you are taking the log of. Is it 4, 4x or 4x^3?
I assume that you mean
log(4x^3) + log(2 x^1/2) = 8
which can be rewritten
log(2^2*x^3) + log 2 + (1/2) log x = 8
or
2log 2 + 3log x + log2 + (1/2) log x = 8
3log2 + (7/2)log x = 8
Use a calculator to solve that. What is the base of your log? It will matter.
To solve the equation log(4x^3) + log(2x^(1/2)) = 8, you can use logarithmic properties to simplify the equation.
Step 1: Combine the logarithms using the product rule of logarithms:
log(4x^3 * 2x^(1/2)) = 8
Step 2: Simplify the expression inside the logarithm:
log(8x^(5/2)) = 8
Step 3: Remove the logarithm by using the inverse function, which is exponentiation, and rewrite the equation:
8x^(5/2) = 10^8
Step 4: Simplify the right side of the equation:
8x^(5/2) = 100,000,000
Step 5: Divide both sides of the equation by 8 to isolate x:
x^(5/2) = 12,500,000
Step 6: Take the square root of both sides of the equation to solve for x:
(sqrt(x^(5/2))) = sqrt(12,500,000)
x^(5/2) = 3,535.53
Step 7: Raise both sides of the equation to the power of 2/5 to solve for x:
x = (3,535.53)^(2/5)
x ≈ 39.71
Therefore, the solution to the equation is x ≈ 39.71.
To solve the given equation log4x^3 + log2x^1/2 = 8, we can use the rules of logarithms to simplify the equation and then solve for x.
Step 1: Apply the logarithmic property loga + logb = log(ab) to combine the two logarithms on the left-hand side:
log4x^3 + log2x^1/2 = log(4x^3 * 2x^1/2)
Step 2: Simplify the expression within the parentheses:
log(4x^3 * 2x^1/2) = log(8x^4)
Step 3: Apply the logarithmic property loga(x) = y if and only if a^y = x to rewrite the equation:
log(8x^4) = 8
Step 4: Convert the equation to exponential form:
8x^4 = 10^8
Step 5: Simplify 10^8:
8x^4 = 100,000,000
Step 6: Divide both sides of the equation by 8 to isolate x^4:
x^4 = 100,000,000 / 8
Step 7: Simplify the right-hand side:
x^4 = 12,500,000
Step 8: Take the fourth root of both sides to solve for x:
x = √(12,500,000)
x ≈ 353.553
Therefore, the solution to the given equation is x ≈ 353.553.